The chi-square distribution is commonly used in hypothesis testing, particularly the chi-squared test for goodness of fit.

The chi-square distribution uses the following parameter.

The probability density function (pdf) is

y=f(xν)=x(ν−2)/2e−x/22ν2Γ(ν/2)

where Γ( · ) is the Gamma function, ν is the degrees of freedom, and x ≥ 0.

The cumulative distribution function (cdf) is

p=F(xν)=x0t(ν−2)/2e−t/22ν/2Γ(ν/2)dt

where Γ( · ) is the Gamma function, ν is the degrees of freedom, and x ≥ 0.

The mean is ν.

The variance is 2ν.

The χ2 distribution is a special case of the gamma distribution where b = 2 in the equation for gamma distribution below.

y=f(xa,b)=1baΓ(a)xa−1exb

The χ2 distribution gets special attention because of its importance in normal sampling theory. If a set of n observations is normally distributed with variance σ2, and s2 is the sample standard deviation, then

(n−1)s2σ2～χ2(n−1)

This relationship is used to calculate confidence intervals for the estimate of the normal parameter σ2 in the function normfit.

Compute the pdf of a chi-square distribution with 4 degrees of freedom.

Plot the pdf.

The chi-square distribution is skewed to the right, especially for few degrees of freedom.